SMOOTH UPPER BOUNDS FOR THE PRICE FUNCTION OF AMERICAN STYLE OPTIONS

Louis Bhim and Reiichiro Kawai ()
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Louis Bhim: Optiver Asia Pacific, 39 Hunter St., Sydney, NSW 2000, Australia
Reiichiro Kawai: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

International Journal of Theoretical and Applied Finance (IJTAF), 2018, vol. 21, issue 01, 1-38

Abstract: We introduce a new approach for systematically obtaining smooth deterministic upper bounds for the price function of American style options. These bounding functions are characterized by sufficient conditions, under which the bounds may be infimized. In a single implementation, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the price function from above over the whole problem domain both in time and state. As a consequence, these global bounds store a continuum of information in the form of a finite number of polynomial coefficients. The proposed approach achieves these bounds, free from statistical error, without recourse to sample path simulation, without truncating the naturally unbounded domain that arises in this problem, and without discretizing the time and state variables. Throughout the paper, we demonstrate the effectiveness of the proposed method in obtaining tight upper bounds for American style option prices in a variety of market models and with various payoff structures, such as the multivariate Black Scholes and Heston stochastic volatility models and the American put and butterfly payoff structures. We also discuss extensions of the proposed methodology to perpetual American style options and frameworks in which the underlying asset contains jumps.

Keywords: American option price; optimal stopping; dual approach; free boundary problem; polynomial bounds (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0219024918500097

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